Multivariable Bohr Inequalities

Operator-valued multivariable Bohr type inequalities are obtained for: (i) a class of noncommutative holomorphic functions on the open unit ball of B(H)n, generalizing the analytic functions on the open unit disc; (ii) the noncommutative disc algebra An and the noncommutative analytic Toeplitz algebra F∞ n ; (iii) a class of noncommutative selfadjoint harmonic functions on the open unit ball of B(H)n, generalizing the real-valued harmonic functions on the open unit disc; (iv) the Cuntz-Toeplitz algebra C∗(S1, . . . , Sn), the reduced (resp. full) group C∗-algebra C∗ red(Fn) (resp. C ∗(Fn)) of the free group with n generators; (v) a class of analytic functions on the open unit ball of Cn. The classical Bohr inequality is shown to be a consequence of Fejér’s inequality for the coefficients of positive trigonometric polynomials and Haagerup-de la Harpe inequality for nilpotent operators. Moreover, we provide an inequality which, for analytic polynomials on the open unit disc, is sharper than Bohr’s inequality. Introduction Let f(z) := ∞ ∑ k=0 akz k be an analytic function on the open unit disc D := {z ∈ C : |z| < 1} such that ‖f‖∞ ≤ 1. Bohr’s inequality [5] asserts that ∞ ∑ k=0 r|ak| ≤ 1 for 0 ≤ r ≤ 1 3 . Originally, the inequality was obtained for 0 ≤ r ≤ 16 . The fact that 1 3 is the best possible constant was obtained independently by M. Riesz, Schur, and Weiner. Other proofs were later obtained by Sidon [25] and Tomic [26]. Dixon [7] used Bohr’s inequality in connection with the long-standing problem of characterizing Banach algebras satisfying the von Neumann inequality [28] (see also [11] and [14]). In recent years, multivariable analogues of Bohr’s inequality were considered by several authors (see [1], [4], [6], and [12]). Paulsen and Singh [13] used positivity methods to obtain operator-valued generalizations of Bohr’s inequality in the single variable case. Received by the editors December 27, 2004 and, in revised form, August 17, 2005. 2000 Mathematics Subject Classification. Primary 47A20, 47A56; Secondary 47A13, 47A63.